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14.10 : The Small x Assumption

If a reaction has a small equilibrium constant, the equilibrium position favors the reactants. In such reactions, a negligible change in concentration may occur if the initial concentrations of reactants are high and the Kc value is small. In such circumstances, the equilibrium concentration is approximately equal to its initial concentration.  This estimation can be used to simplify the equilibrium calculations by assuming that some equilibrium concentrations are equal to the initial concentrations. However, to make this assumption, the change in the concentration of a weak acid or base, i.e., x, must be less than 5% of its initial concentration. If x is more than 5%, then the quadratic formula needs to be used to solve the equilibrium equation.  

Calculation of Equilibrium Concentrations Using an Algebra-Simplifying Assumption

What are the concentrations at equilibrium of a 0.15 M solution of HCN?

Polymerase chain reaction diagram showing DNA amplification process with thermal cycles.

Using x to represent the concentration of each product at equilibrium gives this ICE table.

 HCN (aq)           H+ (aq) CN(aq)
Initial Concentration (M) 0.15 0 0
Change (M) −x +x +x
Equilibrium Concentration (M) 0.15 − x x x

Substitute the equilibrium concentration terms into the Kc expression

Static equilibrium; ΣFx=0, ΣFy=0, ΣM=0; diagram with applied forces and moments; vector analysis.

rearrange to the quadratic form and solve for x

Static equilibrium, ΣFx=0, MA=0, diagram, illustrating force balance on an object in mechanical stability.

Thus, [H+] = [CN] = x = 8.6 × 10–6 M and [HCN] = 0.15 – x = 0.15 M.

Note in this case that the change in concentration is significantly less than the initial concentration (a consequence of the small K), and so the initial concentration experiences a negligible change:

Nernst equation diagram illustrating electrochemical cell potential with ion concentration.

This approximation allows for a more expedient mathematical approach to the calculation that avoids the need to solve for the roots of a quadratic equation:

Chemical equilibrium equation CH3COOH ⇌ CH3COO− + H+. Acid ionization concept diagram.

The value of x calculated is, indeed, much less than the initial concentration

Spectroscope diagram illustrating light dispersion and measurement in optical studies.

and so the approximation was justified. If this simplified approach were to yield a value for x that did not justify the approximation, the calculation would need to be repeated without making the approximation.

This text has been adapted from Openstax, Chemistry 2e, Section 13.4 Equilibrium Calculations.

Tags

Small X AssumptionEquilibrium ExpressionQuadratic FormulaEquilibrium ConstantInitial ConcentrationsReactantsProductsApproximationChange In ConcentrationICE TableCarbonyl DichlorideCarbon MonoxideChlorine Gas

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